Adiabatic Losses and Stochastic Particle Acceleration in Gamma-Ray Burst Blast Waves

Abstract

We treat the problem of adiabatic losses and stochastic particle acceleration in gamma-ray burst (GRB) blast waves that decelerate by sweeping up matter from an external medium. The shocked fluid is assumed to be represented by a homogeneous expanding shell. The energy lost by nonthermal particles through adiabatic expansion is converted to the bulk kinetic energy of the outflow, permitting the evolution of the bulk Lorentz factor Gamma of the blast wave to be self-consistently calculated. The behavior of the system is shown to reproduce the hydrodynamic self-similar solutions in the relativistic and nonrelativistic limits, and the formalism is applicable to scenarios that are intermediate between the adiabatic and fully radiative regimes. Nonthermal particle energization through stochastic gyroresonant acceleration with magnetic turbulence in the blast wave is treated by employing energy-gain rates and diffusive escape timescales based upon expressions derived in the quasilinear regime. If the magnetic field in the shocked fluid approaches its equipartition value, this process can accelerate escaping particles to >~10^{20} eV energies, consistent with the hypothesis that ultra-high energy cosmic rays (UHECRs) are accelerated by GRB blast waves. Due to particle trapping by the magnetic turbulence, only the highest energy particles can escape during the prompt and afterglow phases of a GRB for acceleration by a Kolmogorov spectrum of MHD turbulence. Lower energy particles begin to escape as the blast wave becomes nonrelativistic and shock Fermi acceleration becomes more important.